Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields

Details Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields

1997-11-12

arxiv.org/abs/alg-geom/9711015v2

Mathematics

Algebraic Geometry

LaTeX 2e, 58 pages, revised and extended

Scientific paper

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type $^{3,6}\D_4$, nor $\E_6$. We discuss some consequences derived from these, e.g., by giving some new criteria for weak approximation in algebraic groups over number fields, by indicating a new way to give examples of non stably rational algebraic groups over local fields and application to norm principle.

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